Energy Identities of ADI-FDTD Method with Periodic Structure

In this paper, a new kind of energy identities for the Maxwell equations with periodic boundary conditions is proposed and then proved rigorously by the energy methods. By these identities, several modified energy identities of the ADI-FDTD scheme for the two dimensional (2D) Maxwell equations with the periodic boundary conditions are derived. Also by these identities it is proved that 2D-ADI-FDTD is approximately energy conserved and unconditionally stable in the discrete L ² and H ¹ norms. Experiments are provided and the numerical results confirm the theoretical analysis on stability and energy conservation.

• Stability
• Energy Conservation
• ADI-FDTD
• Maxwell Equations

1. Introduction

The alternative direction implicit finite difference time domain (ADI-FDTD) methods, proposed in [1] [2] , are interesting and efficient methods for numerical solutions of Maxwell equations in time domain, and cause many researchers’ work since ADI-FDTD overcomes the stability constraint of the FDTD scheme [3] . For example, it was proved by Fourier methods in [4] -[8] that the ADI-FDTD methods are unconditionally stable and have reasonable numerical dispersion error; Reference [9] studied the divergence property; Reference [10] studied ADI-FDTD in a perfectly matched medium; Reference [11] gave an efficient PML implementation for the ADI-FDTD method. By Poynting’s theorem, Energy conservation is an important property for Maxwell equations and good numerical method should conform it. In 2012, Gao [12] proposed several new energy identities of the two dimensional (2D) Maxwell equations with the perfectly electric conducting (PEC) boundary conditions and proved that ADI-FDTD is approximately energy conserved and unconditionally in the discrete L² and H¹ norms. Is there any other structure which can keep energy conservation for Maxwell equations? Is there any other energy identity for ADI-FDTD method? This two interesting questions promote us to find other energy- conservation structure.

In this paper, we focus our attention on structure with periodic boundary conditions and propose energy identities in L² and H¹ norms of the 2D Maxwell equations with periodic boundary conditions. We derive the energy identities of ADI-FDTD for the 2D Maxwell equations (2D-ADI-FDTD) with periodic boundary conditions by a new energy method. Several modified energy identities of 2D-ADI-FDTD in terms of the discrete L² and H¹ norms are presented. By these identities it is proved that 2D-ADI-FDTD with the periodic boundary conditions is unconditionally stable and approximately energy conserved under the discrete L² and H¹ norms. To test the analysis, experiments to solve a simple problem with exact solution are provided. Computational results of the energy and error in terms of the discrete L² and H¹ norms confirm the analysis on the energy conservation and the unconditional stability.

The remaining parts of the paper are organized as follows. In Section 2, energy identities of the 2D Maxwell equations with periodic conditions in L² and H¹ norms are first derived. In Section 3, several modified energy identities of the 2D-ADI-FDTD method are derived, the unconditional stability and the approximate energy conservation in the discrete L² and H¹ norms are then proved. In Section 4, the numerical experiments are presented.

2. Energy Conservation of Maxwell Equations and 2D-ADI-FDTD

Consider the two-dimensional (2D) Maxwell equations:

in a rectangular domain with electric permittivity ε and magnetic permeability μ, where ε and μ are positive constants; and denote the electric and magnetic fields, ,.

We assume that the rectangular region Ω is surrounded by periodic boundaries, so the boundary conditions can be written as

We also assume the initial conditions

It can be derived by integration by parts and the periodic boundary conditions (2.2)-(2.3) that the above Maxwell equations have the energy identities:

Lemma 2.1 Let and be the solution of the Maxwellsystems (2.1)-(2.4). Then

where and in what follows, denotes the L² norm with the weights ε (corresponding electric field) or µ (magnetic field). For example,

Identity (2.5) is called the Poynting Theorem and can be seen in many classical physics books. Besides the above energy identities, we found new ones below.

Theorem 2.2 Let and be the solution of the Maxwell systems (2.1)-(2.4), the same as those in Lemma 2.1. Then, the following energy identities hold

where u = x or y, and is the H¹ norm (the H¹ norm of f is defined by, where

, ,. is also called the H¹-semi norm of f).

Proof. First, we prove Equation (2.7) with u = x. Differentiating each of the Equations in (2.1) with respect to x leads to

By the integration by parts and the periodic boundary conditions (2.2)-(2.3), we have

where

Multiplying the Equations (2.9) by, and respectively, integrating both sides over and using (2.10), we have

From (2.1) and the boundary conditions (2.2)-(2.3) we note that

So,. Then, by integrating (2.12) with respect to time over, we get equation (2.7) with u = x. Similarly, the identity (2.7) with u = y can be proved. Combining (2.5) and (2.7) leads to (2.8).

The 2D-ADI-FDTD Scheme

The alternating direction implicit FDTD method for the 2D Maxwell equations (denoted by 2D-ADI-FDTD) was proposed by (Namiki, 1999). For convenience in analysis of this scheme, next we give some notations. Let

where Δx and Δy are the mesh sizes along x and y directions, ∆t is the time step, I, J and N are positive integers. For a grid function, define

where u = x, y or t. For, , , define some discrete energy norms based on the Yee staggered grids (Yee, 1966),

Other norms:, and are similarly defined. Denote by and

the approximations of (u = x, y) and, respectively. Then the 2D-ADI-FDTD scheme for (2.1) is written as

Stage 1:

Stage 2:

For simplicity in notations, we sometimes omit the subscripts of these field values without causing any ambiguity. By the definition of cross product of vectors, the boundary conditions for (2.2)-(2.3) become

 (2.20)

where or. Finally, the initial values and of the2D-ADI-FDTD scheme are obtained by the initial condition (2.4).

3. Modified Energy Identities and Stability of 2D-ADI-FDTD in H 1 Norm

In this Section we derive modified energy identities of 2D-ADI-FDTD and prove its energy conservation and unconditional stability in the discrete H¹ norm.

Theorem 3.1 Let, and be the solution of the ADI-FDTD scheme (2.14)-(2.19). Then the following modified energy identities hold,

where for, and or 0

Proof. First we prove (3.1). Applying to the Equations (2.14)-(2.19), and rearranging the terms by the time levels, we have

Multiplying both sides of the equations, (3.3)-(3.4) by respectively, and those of (3.5) by, and taking the square of the updated equations lead to

Applying summation by parts, we see that

where we have used that and that, which can be obtained from the periodic boundary conditions. Similarly, we get that

So, if summing each of the Equalities (3.9)-(3.11) over their subscripts, adding the updated equations, multiplying both sides by ΔxΔy, and using the two identities, (3.12) and (3.13), together with the norms defined in Subsection 2.2, we arrive at

Similar argument is applied to the second Stage (3.6)-(3.8), we have

Combination of (3.14) and (3.15) leads to the identity (3.1). Identity (3.2) is similarly derived by repeating the above argument from the operated Equations (2.14)-(2.19) by. This completes the proof of Theorem 3.1.

In the above proof, if taking as the identity operator, we obtain that

Theorem 3.2 Let, and be the solution of 2D-ADI-FDTD. Then, the following energy identities hold

Combining the results in Theorems 3.1 and 3.2 we have

Theorem 3.3 If the discrete H¹ semi-norm and H¹ norm of the solution of 2D-ADI-FDTD are denoted respectively by

then, the following energy identities for 2D-ADI-FDTD hold

Remark 3.4 It is easy to see that the identities in Theorems 3.1, 3.2 and 3.3converge to those in Lemma 2.1 and Theorem 2.2 as the discrete step sizes approach zero. This means that2D-ADI-FDTD is approximately energy-conserved and unconditionally stable in the modified discrete form of the L² and H¹ norms.

4. Numerical Experiments

In this section we solve a model problem by 2D-ADI-FDTD, and then test the analysis of the stability and energy conservation in Section 3 by comparing the numerical solution with the exact solution of the model. The model considered is the Maxwell equations (2.1) with, , , and its exact so-

lution is:,

It is easy to compute the norms of this solution are

4.1. Simulation of the Error and Stability

To show the accuracy of 2D-ADI-FDTD, we define the errors:

where, , are the true values of the exact solution. Denote the error and relative error in the norms defined in Section 3 by ErL₂, R-ErL₂, ErH₁ and R-ErH₁, i.e.

where log is the logarithmic function.

Table 1 gives the error and relative error of the numerical solution of the model problem computed by 2D- ADI-FDTD in the norms, and the convergence rates with different time step sizes Δt = 4h, 2h and h, when Δx = Δy = h = 0.01 is fixed and T = 1. From these results we see that the convergence rate of 2D-ADI-FDTD with respect to time is approximately 2 and that 2D-ADI-FDTD is unconditionally stable (when Δt = Δx = Δy = h,

the CFL number).

Table 2 lists the similar results to Table 1 when Δt = 0.1h is fixed, Δx = Δy varies from 2h, h and 0.5h, and the time length T = 1. From the columns “Rate” we see that 2D-ADI-FDTD is of second order in space under the discrete L² and H¹ norm.

4.2. Simulation of the Energy Conservation of 2D-ADI-FDTD

In this subsection we check the energy conservation of 2D-ADI-FDTD by computing the modified energy norms derived in Section 3 for the solution to the scheme. Denote these modified energy norms by

In Table 3 are presented the energy norms of the solution of the 2D-ADI-FDTD scheme at the time levels n = 0, n = 1000 and n = 4000 (the third to fifth rows), and the absolute values of their difference (the last two rows), where the sizes of the spatial and time steps are Δx = Δy = 0.01, Δt = 0.04. The second row shows the four kind of energies of the exact solution computed by using the definitions of. From these value we see that 2D-ADI-FDTD is approximately energy-conserved.

Table 1

Error of in L 2 and H 1 with Δx = Δy = h and different Δt

Δt	R-ErL2	ErL2	Rate	R-ErH1	ErH1	Rate
4h	6.0284e−2	8.5254e−2		6.0287e−2	7.6675e−1
2h	1.6264e−2	2.3001e−2	1.8901	1.6265e−2	2.0595e−1	1.8901
h	5.1571e−3	7.2932e−3	1.6571	5.1571e−3	6.5229e−2	1.3182

Table 2

Error of in L 2 and H 1 with Δt = 0.1h and different spatial step sizes

Δx = Δy	R-ErL2	ErL2	Rate	R-ErH1	ErH1	Rate
2h	5.0019e−3	8.3182e−3		5.0019e−3	7.4333e−3
h	1.4981e−3	2.1186e−3	1.7393	1.4981e−3	1.8942e−3	1.7393
0.5h	4.0200e−4	5.6851e−4	1.8979	4.0200e−4	5.0834e−4	1.8978

Table 3

Energy of and its error when Δx = Δy = h = 0.01, Δt = 4h and n = 0, 1000, 4000

Fields\Norms
	8.9367	8.9367	1.4226	12.7183
	8.9367	8.9367	1.4226	12.7183
	8.9367	8.9367	1.4226	12.7183
	8.9367	8.9367	1.4226	12.7183
	3.2685e−13	3.2685e−13	5.2403e−14	4.6718e−13
	3.2685e−13	3.2685e−13	5.2403e−14	4.6718e−13

5. Conclusion

In this paper, the modified energy identities of the 2D-ADI-FDTD scheme with the periodic boundary conditions in the discrete L² and H¹ norms are established which show that this scheme is approximately energy conserved in terms of the two energy norms. By the deriving methods for the energy identities, new kind of energy identities of the Maxwell equations are proposed and proved by the new energy method. Numerical experiments are provided and confirm the analysis of 2D-ADI-FDTD.

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